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Theorem exmodc 1993
Description: If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
Assertion
Ref Expression
exmodc  |-  (DECID  E. x ph  ->  ( E. x ph  \/  E* x ph ) )

Proof of Theorem exmodc
StepHypRef Expression
1 df-dc 777 . 2  |-  (DECID  E. x ph 
<->  ( E. x ph  \/  -.  E. x ph ) )
2 pm2.21 580 . . . 4  |-  ( -. 
E. x ph  ->  ( E. x ph  ->  E! x ph ) )
3 df-mo 1947 . . . 4  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
42, 3sylibr 132 . . 3  |-  ( -. 
E. x ph  ->  E* x ph )
54orim2i 711 . 2  |-  ( ( E. x ph  \/  -.  E. x ph )  ->  ( E. x ph  \/  E* x ph )
)
61, 5sylbi 119 1  |-  (DECID  E. x ph  ->  ( E. x ph  \/  E* x ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 662  DECID wdc 776   E.wex 1422   E!weu 1943   E*wmo 1944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777  df-mo 1947
This theorem is referenced by: (None)
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